1. Wash, hull, and quarter 1 cup of strawberries. Slice the quarters very thinly. Place in a mixing bowl and pour 1/2 cup of the sugar and the liquor over the berries. Mix, cover, and refrigerate for at least 2 hours, and as long as 2 days.
  2. Wash, hull, and slice the remaining berries, and purée in a blender until very smooth. Strain through a fine-mesh strainer to remove the seeds, then measure out 1 1/2 cups of the strained purée. Reserve any extra for another use.
  3. Whisk the xanthan gum into the remaining 3/4 cup sugar in a large mixing bowl. Whisk in the 1 1/2 cups of purée, half & half, and corn syrup, whisking until fully combined. Add salt to taste and, if the mixture is too sweet, a few drops of fresh lemon juice.
  4. Chill the base until very cold, then freeze in an ice cream maker according to the manufacturer's instructions. In the last minute of churning, retrieve the sliced strawberries from the refrigerator, drain the liquid, and add to the ice cream for the final few turns.
  5. Transfer the ice cream to a covered container and freeze for 2 to 3 hours before serving.